The original proof I heard in high school just assume no common factor between p and q i.e. we can eliminate all factors with common factors for x being rationa

The original proof I heard in high school just assume no common factor between p and q i.e. we can eliminate all factors with common factors for x being rational as m/n to p/q. I miss any argument involving any fundamental theorem. The issue is that it is a contradiction, as it is obvious p and q have 2 as their common factor. This is a proof by contradiction and if logic right, the assumption they are rational is wrong. The historical original by p school (as they are Greek) involving geometry not algebra.